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Sets of Natural Numbers with Proscribed Subsets
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Kevin O'Bryant

Department of Mathematics

College of Staten Island (CUNY)

Staten Island, NY 10314

USA

**Abstract:**

Let 𝒜 be a set of subsets of the natural numbers, and let *G*_{𝒜}(*n*) be the maximum cardinality of a subset of {1, 2, . . . , *n*} that does not have any subsets that are in 𝒜. We consider the general problem of giving upper bounds on *G*_{𝒜}(*n*), and give new results for some 𝒜 that are closed under dilation. We specifically address some examples, including sets that do not contain geometric progressions of length *k* with integer ratio, sets that do not contain geometric progressions of length *k* with rational ratio, and sets of integers that do not contain multiplicative squares, i.e., sets of the form {*a*, *ar*, *as*, *ars*}.

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(Concerned with sequences
A003002
A003003
A003004
A003005
A003022
A003142
A156989
A208746
A259026.)

Received October 18 2014; revised versions received June 12 2015; June 30 2015; July 12 2015.
Published in *Journal of Integer Sequences*, July 16 2015.

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